Tangential representations at isolated fixed points of odd-dimensional $G$-manifolds
Bull. Korean Math. Soc. 2008 Vol. 45, No. 1, 33-37
Published online March 1, 2008
Katsuhiro Komiya
Yamaguchi University
Abstract : Let $G$ be a compact abelian Lie group, and $M$ an odd-dimensional closed smooth $G$-manifold. If the fixed point set $M^G \ne \emptyset$ and $\dim M^G = 0$, then $G$ has a subgroup $H$ with $G/H \cong \mathbb{Z}_2$, the cyclic group of order 2. The tangential representation $\tau_x(M)$ of $G$ at $x \in M^G$ is also regarded as a representation of $H$ by restricted action. We show that the number of fixed points is even, and that the tangential representations at fixed points are pairwise isomorphic as representations of $H$.
Keywords : tangential representations, Smith equivalent, isolated fixed points
MSC numbers : 57S15, 57S17
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